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Overview

In the following notebok we extract elliptic OECS in the Agulhas region from the AVISO dataset. The notebook is structured as follows:

  1. Import data from the file 'AVISO.mat' stored in the folder 'data'.

  2. Define computational parameters (such as the number of cores) and data.

  3. Define spatio-temporal domain.

  4. Interpolate velocity from the (discrete) gridded data.

  5. Hyperbolic OECS:

    • Compute rate of strain tensor $ \mathbf{S}(\mathbf{x}, t) $ over meshgrid.

    • Compute eigenvalues $ s_i(\mathbf{x},t) $ and eigenvectors $ e_i(\mathbf{x},t) $ (i = {1,2}) of rate of strain tensor.

    • Compute the sets of isolated local maxima of $ \mathbf{S}(\mathbf{x}, t) $ (=objective saddle-points)

    • Compute repelling OECS as tensorlines tangent to the eigenvectors $ \mathbf{e_1}(\mathbf{x}) $ of the rate of strain tensor. The tensorlines are launched from isolated local maxima (objective saddle-points). \begin{equation} \dfrac{d}{ds}\mathbf{e}(\mathbf{x}) = \mathbf{e_1}(\mathbf{x}) \end{equation}

    • Compute attracting OECS as tensorlines tangent to the eigenvectors $ \mathbf{e_2}(\mathbf{x}) $ of the rate of strain \begin{equation} \dfrac{d}{ds}\mathbf{e}(\mathbf{x}) = \mathbf{e_2}(\mathbf{x}) \end{equation}

    • Stop the integration of the tensorlines as soon as $ |s_2(\mathbf{x},t)| $ stops to be monotonically decreasing or $ |s_2(\mathbf{x},t)| < d_{hyperbolicity} $, where $ d_{hyperbolicity} $ is a threshold imposed on the rate of attraction/repulsion (=hyperbolicity) of the attracting/repelling (=hyperbolic) OECS.

Import Data

Computational parameters and data

Here we define the computational parameters and the data.

Spatio-temporal domain

Here we define the spatio-temporal domain over which to consider the dynamical system.

Interpolate Velocity

In order to evaluate the velocity field at arbitrary locations and times, we must interpolate the discrete velocity data. The interpolation with respect to time is always linear. The interpolation with respect to space can be chosen to be "cubic" or "linear". In order to favour a smooth velocity field, we interpolate the velocity field in space using a cubic interpolant.

Rate of strain tensor over meshgrid of initial conditions

The rate of strain tensor $ S(\mathbf{x}, t) $ at time $ t $ is computed by iterating over meshgrid. The rate of strain tensor at point $ \mathbf{x} $ at time $ t $ is computed from the gradient of the velocity field by using an auxiliary meshgrid. 'aux_grid' specifies the ratio between the auxiliary grid and the original meshgrid. This parameter is generally chosen to be between $ [\dfrac{1}{5}, \dfrac{1}{10}] $.

Eigenvalues/Eigenvectors of rate of strain tensor

We now compute the properties of the rate of strain tensor 'S' such as the eigenvalues 's1', 's2' and eigenvectors 'eigenv1', 'eigenv2'.

Find objective Saddle-points

Objective saddles coincide with local maxima in the maximum eigenvalue field $ s $ of the rate of strain tensor.

Shrinklines (Repelling OECS)

Repelling LCS can be sought among trajectories of the differential equation:

\begin{equation} \mathbf{x}'_0(s) = \mathbf{e}_1(\mathbf{x}_0;t), \label{eq: shrinklines} \end{equation}

with $ \mathbf{e}_1 $ denoting the eigenvector associated to the weakest eigenvalue $ s_1 $ of $ S(\mathbf{x},t) $. The non-orientable vector field is well defiend away from tensorline singularites (points where $ s_1 = s_2 $). The most repelling shrinklines mark initial positions of repelling OECSs. Repelling OECSs can therefore be located as trajectories of eq. \ref{eq: shrinklines} that have locally the largest averaged $ s_2(\mathbf{x},t) $ among all neighbouring shrinklines.

Stretchlines (Attracting OECS)

Attracting LCS can be sought among trajectories of the differential equation:

\begin{equation} \mathbf{x}'_0 = \mathbf{e}_2(\mathbf{x}_0;t), \label{eq: stretchlines} \end{equation}

with $ \mathbf{e}_2 $ denoting the eigenvector associated to the strongest eigenvalue $ s_2 $ of $ S(\mathbf{x},t) $. The non-orientable vector field is well defiend away from tensorline singularites (points where $ s_1 = s_2 $). The most attracting stretchlines mark initial positions of attracting OECSs. Attracting OECSs can therefore be located as trajectories of eq. \ref{eq: stretchlines} that have locally the smallest averaged $ s_1(\mathbf{x},t) $ among all neighbouring stretchlines.

Advect attracting OECS

Attracting/repelling OECS represent short term attractor/repellors of particles in the flow field. They act as the eulerian counterpart to hyperbolic LCS. Hyperbolic OECS act as a short-term unstable/stable manifold of a saddle point. In order to highlight the validity of the extracted hyperbolic OECS and highlight the predictive power of attracting OECS for short time-intervals, we advect attracting OECSs over 7 days and study their effect on nearby particles. We seed around every objective saddle a circular blob of particle which also gets advected by the flow. As shown in the below figure, the attracting OECS play a crucial role in shaping the deformation of the blobs as they stretch and attract the blob originally centered around the objective saddle.

A comparison with the advected attracting OECS computed with the FastTensorlineComputation reveals identical structures.

References

[1] Serra, M., & Haller, G. (2016). Objective Eulerian coherent structures. Chaos: An Interdisciplinary Journal of Nonlinear Science, 26(5), 053110.